Dirichlet series
L(s) = 1 | + 5.24e5·2-s − 5.01e8·3-s + 2.06e11·4-s + 4.18e12·5-s − 2.63e14·6-s − 3.51e15·7-s + 7.20e16·8-s + 1.12e17·9-s + 2.19e18·10-s + 2.56e19·11-s − 1.03e20·12-s − 1.50e20·13-s − 1.84e21·14-s − 2.09e21·15-s + 2.36e22·16-s + 1.65e23·17-s + 5.89e22·18-s + 3.15e23·19-s + 8.62e23·20-s + 1.76e24·21-s + 1.34e25·22-s + 2.34e25·23-s − 3.61e25·24-s − 1.26e26·25-s − 7.88e25·26-s − 2.12e26·27-s − 7.24e26·28-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.747·3-s + 3/2·4-s + 0.490·5-s − 1.05·6-s − 0.815·7-s + 1.41·8-s + 0.249·9-s + 0.693·10-s + 1.39·11-s − 1.12·12-s − 0.370·13-s − 1.15·14-s − 0.366·15-s + 5/4·16-s + 2.84·17-s + 0.352·18-s + 0.695·19-s + 0.735·20-s + 0.609·21-s + 1.96·22-s + 1.50·23-s − 1.05·24-s − 1.73·25-s − 0.524·26-s − 0.702·27-s − 1.22·28-s + ⋯ |
Functional equation
Invariants
Degree: | \(4\) |
Conductor: | \(4\) = \(2^{2}\) |
Sign: | $1$ |
Analytic conductor: | \(300.772\) |
Root analytic conductor: | \(4.16446\) |
Motivic weight: | \(37\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((4,\ 4,\ (\ :37/2, 37/2),\ 1)\) |
Particular Values
\(L(19)\) | \(\approx\) | \(6.226870658\) |
\(L(\frac12)\) | \(\approx\) | \(6.226870658\) |
\(L(\frac{39}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 2 | $C_1$ | \( ( 1 - p^{18} T )^{2} \) |
good | 3 | $D_{4}$ | \( 1 + 167228936 p T + 21233264503222 p^{8} T^{2} + 167228936 p^{38} T^{3} + p^{74} T^{4} \) |
5 | $D_{4}$ | \( 1 - 33460806876 p^{3} T + 73668826362342135734 p^{9} T^{2} - 33460806876 p^{40} T^{3} + p^{74} T^{4} \) | |
7 | $D_{4}$ | \( 1 + 501811279648208 p T + \)\(70\!\cdots\!98\)\( p^{4} T^{2} + 501811279648208 p^{38} T^{3} + p^{74} T^{4} \) | |
11 | $D_{4}$ | \( 1 - 25659945722560373304 T + \)\(60\!\cdots\!66\)\( p^{3} T^{2} - 25659945722560373304 p^{37} T^{3} + p^{74} T^{4} \) | |
13 | $D_{4}$ | \( 1 + \)\(15\!\cdots\!28\)\( T - \)\(38\!\cdots\!02\)\( p^{2} T^{2} + \)\(15\!\cdots\!28\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
17 | $D_{4}$ | \( 1 - \)\(97\!\cdots\!92\)\( p T + \)\(27\!\cdots\!06\)\( p^{3} T^{2} - \)\(97\!\cdots\!92\)\( p^{38} T^{3} + p^{74} T^{4} \) | |
19 | $D_{4}$ | \( 1 - \)\(31\!\cdots\!00\)\( T + \)\(95\!\cdots\!62\)\( p T^{2} - \)\(31\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
23 | $D_{4}$ | \( 1 - \)\(44\!\cdots\!48\)\( p^{2} T + \)\(11\!\cdots\!18\)\( p^{2} T^{2} - \)\(44\!\cdots\!48\)\( p^{39} T^{3} + p^{74} T^{4} \) | |
29 | $D_{4}$ | \( 1 + \)\(53\!\cdots\!80\)\( p T + \)\(30\!\cdots\!98\)\( p^{2} T^{2} + \)\(53\!\cdots\!80\)\( p^{38} T^{3} + p^{74} T^{4} \) | |
31 | $D_{4}$ | \( 1 - \)\(15\!\cdots\!24\)\( p T + \)\(21\!\cdots\!46\)\( p^{2} T^{2} - \)\(15\!\cdots\!24\)\( p^{38} T^{3} + p^{74} T^{4} \) | |
37 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!56\)\( T + \)\(21\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!56\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
41 | $D_{4}$ | \( 1 + \)\(11\!\cdots\!56\)\( T + \)\(10\!\cdots\!46\)\( T^{2} + \)\(11\!\cdots\!56\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
43 | $D_{4}$ | \( 1 - \)\(10\!\cdots\!52\)\( T + \)\(41\!\cdots\!62\)\( T^{2} - \)\(10\!\cdots\!52\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
47 | $D_{4}$ | \( 1 - \)\(16\!\cdots\!04\)\( T + \)\(13\!\cdots\!78\)\( T^{2} - \)\(16\!\cdots\!04\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
53 | $D_{4}$ | \( 1 - \)\(37\!\cdots\!72\)\( T + \)\(21\!\cdots\!22\)\( T^{2} - \)\(37\!\cdots\!72\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
59 | $D_{4}$ | \( 1 - \)\(12\!\cdots\!60\)\( T + \)\(10\!\cdots\!38\)\( T^{2} - \)\(12\!\cdots\!60\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
61 | $D_{4}$ | \( 1 + \)\(13\!\cdots\!16\)\( T + \)\(23\!\cdots\!06\)\( T^{2} + \)\(13\!\cdots\!16\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
67 | $D_{4}$ | \( 1 - \)\(38\!\cdots\!04\)\( T + \)\(31\!\cdots\!58\)\( T^{2} - \)\(38\!\cdots\!04\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
71 | $D_{4}$ | \( 1 - \)\(14\!\cdots\!24\)\( T + \)\(67\!\cdots\!26\)\( T^{2} - \)\(14\!\cdots\!24\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
73 | $D_{4}$ | \( 1 - \)\(58\!\cdots\!72\)\( T + \)\(13\!\cdots\!02\)\( T^{2} - \)\(58\!\cdots\!72\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
79 | $D_{4}$ | \( 1 + \)\(16\!\cdots\!80\)\( T + \)\(39\!\cdots\!18\)\( T^{2} + \)\(16\!\cdots\!80\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
83 | $D_{4}$ | \( 1 - \)\(65\!\cdots\!12\)\( T + \)\(22\!\cdots\!82\)\( T^{2} - \)\(65\!\cdots\!12\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
89 | $D_{4}$ | \( 1 - \)\(69\!\cdots\!80\)\( T + \)\(91\!\cdots\!58\)\( T^{2} - \)\(69\!\cdots\!80\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
97 | $D_{4}$ | \( 1 - \)\(58\!\cdots\!44\)\( T + \)\(69\!\cdots\!58\)\( T^{2} - \)\(58\!\cdots\!44\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
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Imaginary part of the first few zeros on the critical line
−19.52918676248945698910236127338, −18.92295359697899073731511353687, −17.14710500777290908657244849836, −16.87358798325217309157824296804, −15.84735804575658006118092166087, −14.72920695550650551831062717891, −14.01021100532241977055645552609, −13.08662161694281413678592112086, −11.93759671991908892679954913749, −11.83403183625011824699565345154, −10.24672736015074998865242365012, −9.519283170510364101715421510297, −7.51659767810903556344756728415, −6.61171213408361419346297653220, −5.69411201142775513597962334611, −5.23412276945900515983514671119, −3.72453168370163521021637470905, −3.24531358055698392923568953908, −1.73529952177444601908514277158, −0.844811819902837591930642767390, 0.844811819902837591930642767390, 1.73529952177444601908514277158, 3.24531358055698392923568953908, 3.72453168370163521021637470905, 5.23412276945900515983514671119, 5.69411201142775513597962334611, 6.61171213408361419346297653220, 7.51659767810903556344756728415, 9.519283170510364101715421510297, 10.24672736015074998865242365012, 11.83403183625011824699565345154, 11.93759671991908892679954913749, 13.08662161694281413678592112086, 14.01021100532241977055645552609, 14.72920695550650551831062717891, 15.84735804575658006118092166087, 16.87358798325217309157824296804, 17.14710500777290908657244849836, 18.92295359697899073731511353687, 19.52918676248945698910236127338